OpenStax University Physics/E&M/Alternating-Current Circuits

Chapter 15

Alternating-Current Circuits

AC voltage and current ${\displaystyle v=V_{0}\sin(\omega t-\phi )}$ if ${\displaystyle i=I_{0}\sin \omega t.}$
▭ RMS values ${\displaystyle I_{rms}={\tfrac {I_{0}}{\sqrt {2}}}}$ and ${\displaystyle V_{rms}={\tfrac {V_{0}}{\sqrt {2}}}}$ ▭ Impedance ${\displaystyle V_{0}=I_{0}X}$

▭ Resistor ${\displaystyle V_{0}=I_{0}X_{R},\;\phi =0,}$ where ${\displaystyle X_{R}=R}$
▭ Capacitor ${\displaystyle V_{0}=I_{0}X_{C},\;\phi =-{\tfrac {\pi }{2}},}$ where ${\displaystyle X_{C}={\tfrac {1}{\omega C}}}$ ▭ Inductor ${\displaystyle V_{0}=I_{0}X_{L},\;\phi =+{\tfrac {\pi }{2}},}$ where ${\displaystyle X_{L}=\omega L}$
▭ RLC series circuit ${\displaystyle V_{0}=I_{0}Z}$ where ${\displaystyle Z={\sqrt {R^{2}+\left(X_{L}-X_{C}\right)^{2}}}}$ and ${\displaystyle \phi =\tan ^{-1}{\frac {X_{L}-X_{C}}{R}}}$
▭ Resonant angular frequency ${\displaystyle \omega _{0}={\sqrt {\tfrac {1}{LC}}}}$ ▭ Quality factor ${\displaystyle Q={\tfrac {\omega _{0}}{\Delta \omega }}={\tfrac {\omega _{0}L}{R}}}$
▭ Average power ${\displaystyle P_{ave}={\frac {1}{2}}I_{0}V_{0}\cos \phi =I_{rms}V_{rms}\cos \phi }$, where ${\displaystyle \phi =0}$ for a resistor.
▭ Transformer voltages and currents ${\displaystyle {\tfrac {V_{S}}{V_{P}}}={\tfrac {N_{S}}{N_{P}}}={\tfrac {I_{P}}{I_{S}}}}$

For quiz at QB/d_cp2.15

AC voltage and current ${\displaystyle v=V_{0}\sin(\omega t-\phi )}$ if ${\displaystyle i=I_{0}\sin \omega t.}$
RMS values ${\displaystyle I_{rms}={\tfrac {I_{0}}{\sqrt {2}}}}$ and ${\displaystyle V_{rms}={\tfrac {V_{0}}{\sqrt {2}}}}$
Impedance ${\displaystyle V_{0}=I_{0}X}$
Resistor ${\displaystyle V_{0}=I_{0}X_{R},\;\phi =0,}$ where ${\displaystyle X_{R}=R}$
Capacitor ${\displaystyle V_{0}=I_{0}X_{C},\;\phi =-{\tfrac {\pi }{2}},}$ where ${\displaystyle X_{C}={\tfrac {1}{\omega C}}}$
Inductor ${\displaystyle V_{0}=I_{0}X_{L},\;\phi =+{\tfrac {\pi }{2}},}$ where ${\displaystyle X_{L}=\omega L}$
RLC series circuit ${\displaystyle V_{0}=I_{0}Z}$ where ${\displaystyle Z={\sqrt {R^{2}+\left(X_{L}-X_{C}\right)^{2}}}}$ and ${\displaystyle \phi =\tan ^{-1}{\frac {X_{L}-X_{C}}{R}}}$
Resonant angular frequency ${\displaystyle \omega _{0}={\sqrt {\tfrac {1}{LC}}}}$
Quality factor ${\displaystyle Q={\tfrac {\omega _{0}}{\Delta \omega }}={\tfrac {\omega _{0}L}{R}}}$
Average power ${\displaystyle P_{ave}={\frac {1}{2}}I_{0}V_{0}\cos \phi =I_{rms}V_{rms}\cos \phi }$
Transformer voltages and currents ${\displaystyle {\tfrac {V_{S}}{V_{P}}}={\tfrac {N_{S}}{N_{P}}}={\tfrac {I_{P}}{I_{S}}}}$